Chapter 3. The Geostationary Orbit 第三章 定點衛星通訊軌道 亞洲大學 資訊工程學系碩士班 呂克明教授 二○○六年十月二日 1/1/2019

Chapter 3. The Geostationary Orbit (第三章定點衛星通訊軌道) Introduction (緒言) Antenna Look Angles (天線視角) The Polar Mount Antenna (極座標天線) Limit of Visibility (可見度之極限) Near Geostationary Orbits (近定點衛星的軌道) Earth Eclipse of Satellite (衛星的地蝕現象) Sun Transit Outage (日凌現象) Launching Orbits (發射軌道) Homework problems (習題) 1/1/2019

Introduction (緒言) Three (3) conditions are required for an orbit to be geostationary: The satellite must travel eastward at the same rotational speed the earth. The orbit must be circular. The inclination of the orbit must be zero. Given the period P for the geostationary is 23 h, 56 min, 4 s, or 86,164 seconds (the reciprocal of this is 1.00273896 rev/day) and earth’s equatorial radius, R= 6,378 km, the geostationary height can be derived as 35,786 km. The value is often round up to 36,000 km. In fact, the gravitational fields of the sun and moon produce a shift of about 0.85 degree per year in inclination. Also, the earth’s equatorial elliptically causes the satellite to drift eastward along the orbit. 1/1/2019

Antenna Look Angles (天線視角) When installing an earth station, we need calculate the antenna look angles. Definition: The look angles (視角) for the ground station antenna are the azimuth and elevation angles required at the antenna so that it points directly at satellite. The azimuth angle (方位角), A and elevation angle (俯仰角), El are needed for given the values of latitude and longitude of an earth station antenna. Azimuth angle for an earth station antenna: Equation 3.10 Elevation angle for an earth station antenna: Equation 3.12 Azimuth-elevation angles for an earth station location of given the values of latitude, 48.42 degree N, and longitude, 89.26 degree W. All Ku-band satellites are shown in Fig. 3-4. Do not worry about the formulas. They are built in all Direct PC and Direct TV antenna installation software. 1/1/2019

The Polar Mount Antenna (極座標天線) Where the home antenna has to be steer-able, expense usually precludes (排除) the use of separate (個別的) azimuth and elevation actuators (激勵器, 驅動器). A single actuator is used which moves the antenna in a circular arc (圓弧). This is known as a polar mount antenna. The antenna bore sight (瞄準的方向, 準星或槍膛歸零) is normal to the polar axis. Please see Fig. 3.5a. The dish is tilted at an angle relative to the polar mount until the bore sight is pointing at a satellite position. The angle of tilt, the declination (傾斜角), is calculated in Equation 3.15. 1/1/2019

Limits of Visibility (可見度之極限) There will be east and west limits on the geostationary are visible from any given earth station. The limits will be set by the geographic coordinates (地理座標) of the earth station and the antenna elevation (天線位置的高度). The lowest elevation in theory is zero, when the antenna is pointing along the horizontal. The longitudinal limits can be made by considering an earth at the equator. It is 81.3 degree via Equation 3.16. The limits of visibility will also depend on the earth station latitude (地球站位置的高度) and a typical minimum value of elevation, 5 degree. The angle B is shown in Equation 3.19. Please note that the larger latitude, the smaller angle B. 1/1/2019

Near Geostationary Orbits (近定點衛星的軌道) The perturbing forces that cause an orbit to depart from the ideal keplerian orbit are: the gravitational fields of the moon and the sun the non-spherical shape of the earth the solar radiation pressure The reaction of the satellite itself Two-lines orbital elements are published at regular intervals. The period for a geostationary satellite is 23 h, 56 min, 4 s or 86,164 s. The reciprocal of this is 1.00273896 rev/day. The term geosynchronous satellite (同步衛星) is used in many cases instead of geostationary (定點衛星) to describe these near-geostationary satellites. Please note that the geosynchronous satellite does not have to be near-geostationary. Why? 1/1/2019

Earth Eclipse of Satellite (衛星的地蝕現象) If the earth’s equatorial plane coincided with the plane of the orbit around the sun, geostationary satellites would be eclipsed by the earth once each day (由於同步衛星的軌道面和赤道面在同一平面上，理論上，每一天會發生一次衛星-地球-太陽在一條線上的地球遮掩現象，或稱地蝕現象。) The equinox (晝夜平分點) and the solstice (至日,最高點,白晝最長和最短之日分別是夏至和冬至) The spring equinox (春分) is the first day of spring, and the autumnal equinox (秋分) is the first day of autumn. Eclipse begin 23 days before equinox and end 23 days after equinox. The eclipse lasts about 10 min at the beginning and end of the eclipse period and increases to a maximum duration of about 72 minutes at full eclipse. During the eclipse, the solar cells do not function, and operating power must be supplied from batteries. (當衛星-地球-太陽在一條線上的最短10分鐘到最長72分鐘的時段內，因為沒有太陽光的充電，衛星操作的能源是從蓄電池來的。這種現象發生的時候，衛星的太陽能電池無法充電,但是衛星依然可以工作。) 1/1/2019

Sun Transit Outage (日凌現象) During the equinoxes, the sun comes within the beam-width of the earth station antenna (地球-衛星-太陽在一條線上). Sun transit outage: When this happens, the sun appears as an extremely noisy source which completely blanks out the signal from the satellite. This effect is termed sun transit outage. (太陽通過發生衛星故障的太陽凌駕現象，或稱日凌現象). It lasts for short periods each day for about 6 days around the equinoxes (還好僅有春分和秋分前後的六天，衛星不能通訊). The occurrence and duration of the sun transit outage depends on the latitude of the earth station, a maximum outage time of 10 min being typical.(每一次僅有十分鐘) 1/1/2019

Launching Orbits (發射軌道) Low-altitude orbit: Satellite may be directly injected into low-altitude orbits, up to about 200 km altitude, from a launch vehicle. Hohmann transfer orbit: The transfer orbit is selected to minimize the energy required for transfer, and such orbit is known as a Hohmann transfer orbit that was discovered in 1925 in his book <The Attainability of Celestial Bodies> written by a West German scientist Walter Hohmann (1880-1945), born at a small town Odenwald, 32 years before human entering space (10/4/1957, Russian Sputnik I, Sputnik 是俄語的”衛星”之義). The Hohmann elliptical orbit is seen to be tangent to the low-altitude orbit at perigee and to the high-altitude orbit at apogee. Velocity changing in the same plane change the geometry of the orbit but not its inclination. In order to change the inclination, a velocity change is required normal to the orbital plane. It can be shown that the smallest inclination obtainable at initial launch is equal to the latitude of the launch site. Thus the farther away from the equator a launch site is, the less useful it is, since the satellite has to carry extra fuel to effect a change in inclination. 1/1/2019

Homework Problems-I (習題) 3.1 Explain what is meant by the geostationary orbit. How do the geostationary orbit and a geosynchronous orbit differ? 3.6 An earth station is located at latitude 35 degree N and longitude 100 degree W. Calculate the antenna look angle for a satellite at 67 degree W. 3.7 An earth station is located at latitude 12 degree S and longitude 52 degree W. Calculate the antenna look angle for a satellite at 70 degree W. 3.8 An earth station is located at latitude 35 degree N and longitude 65 degree E. Calculate the antenna look angle for a satellite at 19 degree E. 3.9 An earth station is located at latitude 30 degree S and longitude 130 degree E. Calculate the antenna look angle for a satellite at 156 degree E. 1/1/2019

Homework Problems- II (習題) 3.19 Explain what is meant by the earth eclipse of an earth-orbiting satellite. 3.20 Explain briefly what is meant by sun transit outage. 1/1/2019